Problem: $ B = \left[\begin{array}{rrr}5 & 4 & 3 \\ 2 & 2 & 2\end{array}\right]$ $ A = \left[\begin{array}{rr}4 & 3 \\ 1 & 2 \\ -1 & 5\end{array}\right]$ What is $ B A$ ?
Answer: Because $ B$ has dimensions $(2\times3)$ and $ A$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ B A = \left[\begin{array}{rrr}{5} & {4} & {3} \\ {2} & {2} & {2}\end{array}\right] \left[\begin{array}{rr}{4} & \color{#DF0030}{3} \\ {1} & \color{#DF0030}{2} \\ {-1} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rr}{5}\cdot{4}+{4}\cdot{1}+{3}\cdot{-1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{4}+{4}\cdot{1}+{3}\cdot{-1} & ? \\ {2}\cdot{4}+{2}\cdot{1}+{2}\cdot{-1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{4}+{4}\cdot{1}+{3}\cdot{-1} & {5}\cdot\color{#DF0030}{3}+{4}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{5} \\ {2}\cdot{4}+{2}\cdot{1}+{2}\cdot{-1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{5}\cdot{4}+{4}\cdot{1}+{3}\cdot{-1} & {5}\cdot\color{#DF0030}{3}+{4}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{5} \\ {2}\cdot{4}+{2}\cdot{1}+{2}\cdot{-1} & {2}\cdot\color{#DF0030}{3}+{2}\cdot\color{#DF0030}{2}+{2}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}21 & 38 \\ 8 & 20\end{array}\right] $